Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in PYTHON3):

Solution:
N = int(input())
P = list(map(float,input().split()))

#dp[i][j] : 最初のi枚のコインを投げた時に、表がj枚となる確率
dp = [[0]*(N+1) for _ in range(N+1)]
dp[0][0] = 1
for i in range(N):
    for j in range(N):
        dp[i+1][j+1] += dp[i][j]*P[i]
        dp[i+1][j] += dp[i][j]*(1-P[i])

ans = sum(dp[N][(N+1)//2:])
print(ans)